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arXiv:0807.3650 · Journal of Computational and Theoretical Nanoscience

Unitary Reflection Groups for Quantum Fault Tolerance

Michel R. P. Planat, M. Kibler

Published Jul 21, 2010• 12 citations
arXivHTMLPDF

Abstract

This paper explores the representation of quantum computing in terms of unitary reflections (unitary transformations that leave invariant a hyperplane of a vector space). The symmetries of qubit systems are found to be supported by Euclidean real reflections (i.e., Coxeter groups) or by specific imprimitive reflection groups, introduced (but not named) in a recent paper [Planat M and Jorrand Ph 2008, {\it J Phys A: Math Theor} {\bf 41}, 182001]. The automorphisms of multiple qubit systems are found to relate to some Clifford operations once the corresponding group of reflections is identified. For a short list, one may point out the Coxeter systems of type $B_3$ and $G_2$ (for single qubits), $D_5$ and $A_4$ (for two qubits), $E_7$ and $E_6$ (for three qubits), the complex reflection groups $G(2^l,2,5)$ and groups No 9 and 31 in the Shephard-Todd list. The relevant fault tolerant subsets of the Clifford groups (the Bell groups) are generated by the Hadamard gate, the $π/4$ phase gate and an entangling (braid) gate [Kauffman L H and Lomonaco S J 2004 {\it New J. of Phys.} {\bf 6}, 134]. Links to the topological view of quantum computing, the lattice approach and the geometry of smooth cubic surfaces are discussed.

DOI
10.1166/jctn.2010.1541

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This paper explores the representation of quantum computing in terms of unitary reflections (unitary transformations that leave invariant a hyperplane of a vector space). The symmetries of qubit systems are found to be supported by Euclidean real reflections (i.e., Coxeter groups) or by specific imprimitive reflection groups, introduced (but not named) in a recent paper [Planat M and Jorrand Ph 2008, {\it J Phys A: Math Theor} {\bf 41}, 182001]. The automorphisms of multiple qubit systems are found to relate to some Clifford operations once the corresponding group of reflections is identified. For a short list, one may point out the Coxeter systems of type B3B_3B3​ and G2G_2G2​ (for single qubits), D5D_5D5​ and A4A_4A4​ (for two qubits), E7E_7E7​ and E6E_6E6​ (for three qubits), the complex reflection groups G(2l,2,5)G(2^l,2,5)G(2l,2,5) and groups No 9 and 31 in the Shephard-Todd list. The relevant fault tolerant subsets of the Clifford groups (the Bell groups) are generated by the Hadamard gate, the π/4π/4π/4 phase gate and an entangling (braid) gate [Kauffman L H and Lomonaco S J 2004 {\it New J. of Phys.} {\bf 6}, 134]. Links to the topological view of quantum computing, the lattice approach and the geometry of smooth cubic surfaces are discussed.
The recent proposal (M Planat and M Kibler, Preprint 0807.3650 [quantph]) of representing Clifford quantum gates in terms of unitary reflections is revisited. In this essay, the geometry of a Clifford group G is expressed as a BN-pair, i.e. a pair of subgroups B and N that generate G, is such that intersection H = B \cap N is normal in G, the group W = N/H is a Coxeter group and two extra axioms are satisfied by the double cosets acting on B. The BN-pair used in this decomposition relies on the swap and match gates already introduced for classically simulating quantum circuits (R Jozsa and A Miyake, Preprint arXiv:0804.4050 [quant-ph]). The two- and three-qubit steps are related to the configuration with 27 lines on a smooth cubic surface.
Peres/Mermin arguments about no-hidden variables in quantum mechanics are used for displaying a pair (R, S) of entangling Clifford quantum gates, acting on two qubits. From them, a natural unitary representation of Coxeter/Weyl groups W(D5) and W(F4) emerges, which is also reflected into the splitting of the n-qubit Clifford group Cn into dipoles C±\pm±n . The union of the three-qubit real Clifford group C+ 3 and the Toffoli gate ensures a orthogonal representation of the Weyl/Coxeter group W(E8), and of its relatives. Other concepts involved are complex reflection groups, BN pairs, unitary group designs and entangled states of the GHZ family.
The first theme of this book-quantum error correction-has now been pre­ sented. We carry on in this chapter with a discussion of fault-tolerant quantum computing (FTQC), which is our second theme. The discussion will take place in the context of quantum stabilizer codes and is based on Refs. [1-6]. See also Refs. [7,8]. Section 5.1 introduces the idea of fault-tolerance and makes ba­ sic definitions, while Section 5.2 presents a procedure for doing fault-tolerant quantum error correction. The theory of FTQC is then taken up in the fol­ lowing five sections. Section 5.3 examines the action of unitary operators on a quantum stabilizer code. It introduces two important groups: (i) the normalizer of the stabilizer S whose elements are encoded unitary operations; and (ii) the Clifford group which is generated by the Hadamard, phase, and CNOT gates. The discussion focuses on unitary operations that belong to the inter­ section of these two groups. By combining measurements on ancilla qubits (Section 5.4) with a four-qubit unitary operation introduced in Section 5.5, it will be possible to fault-tolerantly implement these gates on encoded data for stabilizer codes that encode a single qubit. Section 5.6 presents an extension of this approach that works for stabilizer codes encoding multiple qubits, and Section 5.7 completes the theoretical development by showing how a faulttolerant encoded Toffoli gate can be constructed for any quantum stabilizer code. By this point a theoretical framework is in place that can construct a set of fault-tolerant encoded quantum gates that is capable of universal quan­ tum computation using any quantum stabilizer code. The final two sections illustrate the theory by applying it to the [5,1,3] and [4,2,2] stabilizer codes. A brief summary of universal sets of quantum gates is given in Appendix C.
The term Clifford group was introduced in 1998 by D. Gottesmann in his investigation of quantum error-correcting codes. The simplest Clifford group in multiqubit quantum computation is generated by a restricted set of unitary Clifford gates - the Hadamard, π /4-phase and controlled-X gates. Because of this restriction the Clifford model of quantum computation can be efficiently simulated on a classical computer (the Gottesmann-Knill theorem). However, this fact does not diminish the importance of the Clifford model, since it may serve as a suitable starting point for a full-fledged quantum computation. In the general case of a single or composite quantum system with finite-dimensional Hilbert space the finite Weyl-Heisenberg group of unitary operators defines the quantum kinematics and the states of the quantum register. Then the corresponding Clifford group is defined as the group of unitary operators leaving the Weyl-Heisenberg group invariant. The aim of this contribution is to show that our comprehensive results on symmetries of the Pauli gradings of quantum operator algebras - covering any single as well as composite finite quantum systems - directly correspond to Clifford groups defined as quotients with respect to U(1).
On d´efend la th`ese que les groupes de r´eflections unitaires (les transformations unitaires qui conservent un hyperplan d'un espace vectoriel) assurent les corrections d'erreurs issues du groupe de Pauli Pn `a n qubits, constitu´e des produits tensoriels des matrices de Pauli i, i = 0..3, avec les phases ±1 ou ±i. Le groupe de Clifford Cn = nU 2 U(2n)|UPnU−1 = Pno normalise le groupe de Pauli dans le groupe unitaire. Ce groupe est apparu dans le contexte de certains codes correcteurs d'erreurs autoduaux (E Bannai 99, G Nebe 00), et il est bien connu que C1 et C2 rel`event de la s´equence de Shephard-Todd des groupes de r´eflections complexes. G¨ottesman `a montr´e en 1997 que trois portes suffisent `a g´en´erer Cn: les portes de Hadamard et de phase /4 (`a un qubit) et la porte CNOT (`a deux qubits). Dans ce travail [1] on met aussi en avant certains groupes cristallographiques de r´eflexions sur l'espace euclidien r´eel (groupes de Coxeter), ainsi que les groupes de r´eflections complexes imprimitifs G(2l, 2, 5). Les groupes imprimitifs sont li´es aux bases mutuellement d´ecorr´el´ees de Pn. Les groupes de Coxeter du type B3 et G2 correspondent aux automorphismes de P1 et les syst`emes de Coxeter du type D5 et E6 apparaˆıssent comme une composante d'intrication topologique de C2 et C3. La g´eom´etrie tient donc de la surface cubique de Clebsch r´egl´ee par 27 lignes droites. On montre ´egalement que le groupe C3 contient les symm´etries des octonions via ses uniques sous-groupes d'ordre 168, 6048 et 12096, isomorphes aux groupes de Lie PSL(2, 7), SU(3, 3) et G2(2), dont on donne une nouvelle repr´esentation.